1 The Potential Energy Surface in Molecular Quantum Mechanics
37
which has oscillator eigenvalues and eigenfunctions
E n (X) = E
0
n +
1 −
a 2
4
X
2 ,
ϕ
x
n
(1.66)
where x = x −
aX
2 , the {ϕ n } are the usual harmonic oscillator eigenfunctions and
E 0
n is the energy of the free oscillator 2(n +
1
2 ). For fixed n the spectrum σ (X) is
discrete and, as a function of the X parameters, would be conventionally interpreted
as a ‘potential energy curve’. As far as (1.58) is concerned, ˆ
K o , evaluated at some
point X 0 cannot be regarded as an ‘approximation’ to ˆ
H o , since obviously
ˆ
K o (X 0 ), ˆ
H 1
= 0
so they can be simultaneously diagonalized, and ˆ
H 1 has purely continuous spectrum
(free motion). So we have to consider X in the full problem in its operator form, ˆ
X.
We make the same unitary transformation of the ˆ
x, ˆ
p variables in ˆ
H o as before,
and it is still brought to diagonal form; however ˆ
H 1 will be modified because ˆ
P is
also translated by the operator ˆ
U in (1.65) that generates the coordinate displacement
(cf. (1.60)); thus
ˆ
U
−1 ˆ
P ˆ
U = ˆ
P +
1
2
a ˆ
p
so the transform of ˆ
H 1 contains linear and quadratic terms in ˆ
p.
Nevertheless (1.61) is still valid, and formally we may write ˆ
H 0 as a direct integral
ˆ
H o =
⊕
X
ˆ
K(ˆ x, ˆ
p, X) o dX.
(1.67)
The Schrödinger equation for ˆ
H o in position representation is now an equation involving functions of x and X
ˆ
H o Φ ε
x
, X
= εΦ ε
x
, X
.
(1.68)
Just as before (see (1.54)) the direct integral decomposition (1.67) implies that the
spectrum is purely continuous, explicitly
σ ( ˆ
H o ) =
X
σ (X) = [1, ∞).
(1.69)
ε in (1.68) takes all positive values ≥ 1, where 1 is the minimum value of the oscillator eigenvalue 2(n +
1
2 ). The associated continuum eigenfunctions {Φ} may
formally be written as products of oscillator eigenfunctions (in x ), and delta functions (in X). They don’t lie in Hilbert space of course and one needs the Gel’fand
construction of a rigged space to make sense of the formal calculation [95]. If one
returns to the x variable, the {ϕ n } are functions of x and X, since x = x (X).
37
which has oscillator eigenvalues and eigenfunctions
E n (X) = E
0
n +
1 −
a 2
4
X
2 ,
ϕ
x
n
(1.66)
where x = x −
aX
2 , the {ϕ n } are the usual harmonic oscillator eigenfunctions and
E 0
n is the energy of the free oscillator 2(n +
1
2 ). For fixed n the spectrum σ (X) is
discrete and, as a function of the X parameters, would be conventionally interpreted
as a ‘potential energy curve’. As far as (1.58) is concerned, ˆ
K o , evaluated at some
point X 0 cannot be regarded as an ‘approximation’ to ˆ
H o , since obviously
ˆ
K o (X 0 ), ˆ
H 1
= 0
so they can be simultaneously diagonalized, and ˆ
H 1 has purely continuous spectrum
(free motion). So we have to consider X in the full problem in its operator form, ˆ
X.
We make the same unitary transformation of the ˆ
x, ˆ
p variables in ˆ
H o as before,
and it is still brought to diagonal form; however ˆ
H 1 will be modified because ˆ
P is
also translated by the operator ˆ
U in (1.65) that generates the coordinate displacement
(cf. (1.60)); thus
ˆ
U
−1 ˆ
P ˆ
U = ˆ
P +
1
2
a ˆ
p
so the transform of ˆ
H 1 contains linear and quadratic terms in ˆ
p.
Nevertheless (1.61) is still valid, and formally we may write ˆ
H 0 as a direct integral
ˆ
H o =
⊕
X
ˆ
K(ˆ x, ˆ
p, X) o dX.
(1.67)
The Schrödinger equation for ˆ
H o in position representation is now an equation involving functions of x and X
ˆ
H o Φ ε
x
, X
= εΦ ε
x
, X
.
(1.68)
Just as before (see (1.54)) the direct integral decomposition (1.67) implies that the
spectrum is purely continuous, explicitly
σ ( ˆ
H o ) =
X
σ (X) = [1, ∞).
(1.69)
ε in (1.68) takes all positive values ≥ 1, where 1 is the minimum value of the oscillator eigenvalue 2(n +
1
2 ). The associated continuum eigenfunctions {Φ} may
formally be written as products of oscillator eigenfunctions (in x ), and delta functions (in X). They don’t lie in Hilbert space of course and one needs the Gel’fand
construction of a rigged space to make sense of the formal calculation [95]. If one
returns to the x variable, the {ϕ n } are functions of x and X, since x = x (X).